Using Expert Opinions in Bayesian Prediction of Component Lifetimes in a Shock Model

Abstract

This paper is concerned with the combination of k expert opinions about the lifetimes of n components of a binary system. This problem has been treated in the single component case by Huseby (Huseby, A. B. 1986. Combining experts' opinions, a retrospective approach. Tech. Rep., Centre for Industrial Research, Oslo; Huseby, A. B. 1988. Combining opinions in a predictive case. J. M. Bernardo, M. H. DeGroot, D. V. Lindley, A. F. M. Smith, eds. Bayesian Statistics, Vol. 3. Oxford University Press, NY, 641–651.). Since the experts often share data, he argues that their assessments Will typically be dependent and that this difficulty cannot be handled without making judgements concerning the underlying sources of information and to what extent these are available to each of the experts. The information available to the experts is modeled as a set of observations Y1, …, Ym. These observations are then reconstructed as far as possible from the information provided by the experts and used as a basis for the combined judgement. This is called the retrospective approach. Huseby (Huseby, A. B. 1988. Combining opinions in a predictive case. J. M. Bernardo, M. H. DeGroot, D. V. Lindley, A. F. M. Smith, eds. Bayesian Statistics, Vol. 3. Oxford University Press, NY, 641–651.) treats a predictive case where the uncertain quantity is modeled as a future observation Y, from the same distribution as the Yi's. For the case n > 1, where each expert is giving opinions about more than one component, additional dependencies between the reliabilities of the components come into play. This is for instance true if two or more components are of similar type, are sharing a common environment or are exposed to common cause failures. For the case n = 2, the retrospective approach and, in particular, the predictive case are treated in Natvig (Natvig, B. 1993. Using expert opinions in Bayesian estimation of system reliability. R. E. Barlow, C. A. Clarotti, F. Spizzichiao, eds. Reliability and Decision Making. Elsevier, London, 347–362.). In the present paper, the predictive case is considered for an arbitrary n and for an arbitrary overlapping of the observation sets from the different experts. The component lifetimes are assumed to have a multivariate exponential distribution of the Marshall-Olkin type. At the end of the paper, it is shown how the joint distribution of the lifetimes of the n components can easily be updated in the case of getting real data.